Gap Eigenvalues and Asymptotic Dynamics of Geometric Wave Equations on Hyperbolic Space
@article{Lawrie2015GapEA, title={Gap Eigenvalues and Asymptotic Dynamics of Geometric Wave Equations on Hyperbolic Space}, author={Andrew Lawrie and Sung-Jin Oh and Sohrab Shahshahani}, journal={arXiv: Analysis of PDEs}, year={2015} }
16 Citations
Equivariant Wave Maps on the Hyperbolic Plane with Large Energy
- Mathematics
- 2015
In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in [13, 14]. When the target is the hyperbolic plane…
Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case.
- Mathematics
- 2017
In this paper, we prove that the small energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may…
Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution
- MathematicsCommunications on Pure and Applied Mathematics
- 2021
We consider the Cauchy problem for the Schr\"odinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the…
The Cauchy problem for wave maps on hyperbolic space in dimensions $d \geq 4$
- Mathematics
- 2015
We establish global well-posedness and scattering for wave maps from $d$-dimensional hyperbolic space into Riemannian manifolds of bounded geometry for initial data that is small in the critical…
On Global Dynamics of Schrödinger Map Flows on Hyperbolic Planes Near Harmonic Maps
- MathematicsCommunications in Mathematical Physics
- 2022
The results of this paper are twofold: In the first part, we prove that for Schrodinger map flows from hyperbolic planes to Riemannian surfaces with non-positive sectional curvatures, the harmonic…
Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces
- MathematicsDiscrete & Continuous Dynamical Systems - A
- 2019
In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that…
Equivariant Schr\"odinger maps from two dimensional hyperbolic space
- Mathematics
- 2017
In this article, we consider the equivariant Schr\"odinger map from $\Bbb H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the origin and spatial infinity of the hyperbolic…
Asymptotic behaviors for nonlinear dispersive equations with damping or dissipative terms
- Mathematics
- 2017
In this note, we will review our recent work on the asymptotic behaviors of nonlinear Klein-Gordon equation with damping terms and LandauLifschitz flows from Eucliedean spaces and hyperbolic spaces.…
Asymptotic stability of small energy harmonic maps under the wave map on 2D hyperbolic space
- Mathematics
- 2017
In this paper, we prove that the small energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map. This result may be seen as an example supporting the soliton…
References
SHOWING 1-10 OF 15 REFERENCES
Stability of stationary equivariant wave maps from the hyperbolic plane
- Mathematics
- 2014
In this paper we initiate the study of equivariant wave maps from $2d$ hyperbolic space, ${\Bbb H}^2$, into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than…
On the formation of singularities in the critical $O(3)$ $\sigma$-model
- Mathematics
- 2006
We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from ℝ 2+1 Minkowski space into the sphere S 2 . We establish rigorously and…
Renormalization and blow up for charge one equivariant critical wave maps
- Mathematics
- 2008
We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)$ where u is the polar angle on the…
Relaxation of solitons in nonlinear Schrödinger equations with potential
- Mathematics, Physics
- 2006
Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations
- Physics, Mathematics
- 1999
Abstract. We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a…
On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations
- Mathematics
- 2007
AbstractWe consider nonlinear Schrödinger equations
$$iu_t +\Delta u +\beta (|u|^2)u=0\, ,\, \text{for} (t,x)\in \mathbb{R}\times \mathbb{R}^d,$$ where d ≥ 3 and β is smooth. We prove that symmetric…
On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential
- Mathematics
- 2009
In this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under suitable smoothness and decay assumptions on the potential and a genericity assumption on…
Equivariant wave maps in two space dimensions
- Mathematics
- 2003
Singularities of corotational wave maps from (1 + 2)‐dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 𝕊2…
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
- Mathematics
- 2009
We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We…